Associated Polynomials and Uniform Methods for the Solution of Linear Problems

Solution of Linear Equations with Rational Toeplitz Matrices

“…In both cases, a power series with a nonzero coefficient of z° is invertible in the ring; this corresponds to the fact that an invertible triangular Toeplitz matrix has a triangular Toeplitz inverse. (In the finite case, Traub [8] has given an expression for the (Toeplitz) inverse of a triangular Toeplitz matrix.) Applying the isomorphisms to the power series equations (4a) and (4b) and taking the first « + 1 rows and columns of the corresponding matrix products, starting at the northwest and southeast corners, respectively, and combining the lower and upper triangular Toeplitz matrices gives the desired representation of Tn.…”

“…Since Tn is Toeplitz, £"(d") is lower triangular, and Unihn) is upper triangular, the first equality in (8) shows that for n > 1, /? "_, is the « by « principal submatrix of Rn.…”

Computation of Hermite polynomials

“…In the next section, such a method is developed. Since that procedure determines the inverse of the matrix A, it may be applicable to general Vandermonde matrices (see Traub [13]). …”

Solution of Vandermonde systems of equations

“…In [1] an algorithm was derived for solving a Vandermonde system of equations These systems, and the corresponding dual systems (3) Kra = f appear naturally, and have to be solved, in many applications. Some important examples are interpolation, construction of spline functions [3], approximation of linear functionals [1], [5], etc. In this paper a new algorithm for solving (1) will be developed which is faster, needs less storage and, from experimental results, is more accurate than that in [1].…”

“…This will, however, be an «3-process. Since in [5] Traub has given an algorithm which computes V~x in (l/2)n(7n -9)M and (5/2)n(n -l)A, this is clearly inefficient.…”